How do I adjust an image on a sphere?
From your screen it seems to me you use Sphere object (not IcoSphere).
Blender basic objects are unwrapped automatically. Image node is also automatically mapped with UV coordinates.

For the country flag you can use UVmap generated by default (when object is added).

Switch to UV Editing layout, in 3D view switch to edit mode, select all vertices
in UV Editor appears UVmap so just select all and stretch on X axis until you see image as circle


To avoid repeating texture from backside change Repeat to Extend in Texture node.

Your reference image deformation in corners is same so I guess it was unwrapped in the same way. If you want to avoid thouse deformations you would have to apply black parts separately or change UVmap.
Another option is to use object to control texture coordinates.
If you add any object (I used Empty) and for Sphere material you add some image (logo in my case) you can mix them together.


By default image texture is positioned to object by left up corner, so I used Mapping node to shift center of logo to center of Empty.
To follow Empty surface of Sphere enable Snapping &gt; Face and check Align Rotation to Target.